In particular, we can exploit the Longo-Witten endomorphisms found in [1]}.
Recall that, for an inner symmetric function \(\varphi (z)\) , namely the boundary value on \({\mathbb {R}}\) 
of a bounded analytic function on \(0 < \Im z < \pi \) , one considers the operator
\(V_\varphi := \Gamma (\varphi (P_1))\) , where \(P_1\)  is the generator of the restriction of \(U_{\mathrm {r}}^+\) 
on the one-particle space \({\mathcal {H}}_1\) , \(\varphi (P_1)\) 
denotes a functional calculus and \(\Gamma \)  is the second quantization
(note that in general the order of second quantization and functional calculus cannot be
exchanged: \(\Gamma (\varphi (P_1)) \ne \varphi (\Gamma (P_1))\) .
Then \({\hbox{\rm Ad\,}}V_\varphi \)  preserves \(\mathcal {M}_{\mathrm {r}}\)  and \(V_\varphi \)  commutes with \(U_{\mathrm {r}}^+\) .
Furthermore, \(V_\varphi \)  commutes with \(U_{\mathrm {r}}^-\)  since \(U_{\mathrm {r}}^-(a) = \Gamma \left(\exp \left(\frac{it}{P_1}\right)\right)\) ,
as we see in [2]}.
